FULLY CONVOLUTIONAL APPROACH FOR SIMULATING WAVE DYNAMICS

Abstract

We investigate the performance of fully convolutional networks to predict the motion and interaction of surface waves in open and closed complex geometries. We focus on a U-Net type architecture and assess its ability to capture and extrapolate wave propagation in time as well as the reflection, interference and diffraction of waves. We investigate how well the network generalises both to long-time predictions and to geometric configurations not seen during training. We demonstrate that this neural network is capable of accurately predicting the height distribution of waves on a liquid surface within curved and multi-faceted open and closed geometries, when only simple box and right-angled corner geometries were seen during training. We found that the RMSE of the predictions remained of order 1 × 10 -4 times the characteristic length of the domain for at least 20 time-steps.

1. INTRODUCTION

Predicting the spatio-temporal dynamics of physical systems is a recurrent problem in many areas of science and engineering. A well-established process consists of describing the physical phenomena by human-engineered mathematical models, which capture our current understanding of the physical laws governing the systems, but whose complexity may prevent finding analytical solutions. Scientists therefore frequently turn to numerical solvers to simulate such mathematical models and generate accurate approximations to their solution. The huge progress in machine learning (ML) algorithms and increased availability of computational power during the last decade has motivated a significant growth in the popularity of data-driven physics. In this field, the interpolation capabilities of neural networks (NNs) have been mostly used in two ways: first, to solve partial differential equations (PDEs) in an unsupervised manner (Dissanayake & Phan-Thien, 1994; Lagaris et al., 1998; 2000; Raissi et al., 2019) and second, to predict the physical dynamics from previous observations without knowledge of the underlying equations (Guo et al., 2016; Farimani et al., 2017; Thuerey et al., 2018; Lee & You, 2019) . Unlike the first approach, the latter does not lead to an analytical representation of the physical dynamics, however, it makes feasible to produce predictions for a diversity of physical domains, boundary conditions and initial conditions without needing to re-train the network, provided that the physical laws are unaltered. Recent studies applying convolutional neural networks (CNNs) to simulate fluid dynamics have reported a speed-up of up to four orders of magnitude, in comparison to traditional numerical solvers, while keeping a similar accuracy (Guo et al., 2016) . The major shortcoming of NNs are their often poor generalization to unseen configurations and poor long-time predictions in unsteady simulations.



Figure 1: Rollouts of our U-Net. It simulates wave motion on a fluid surface with the possible existence of solid walls [video].

